Let us start with a theory of the genesis of the idea of reincarnation.
Consider the maladaptive mental habits and thought patterns, the mental imprints, the karmas that we all carry. In some cases these can be traced to experiences in this life, but in other cases they cannot. Much of the project of modern therapy is to uncover through various means the deeply latent memories underlying these karmas, and to thus dissolve them at their source.
But what of those imprints that cannot be matched to anything in your past experience? Your absolutely random phobia of spiders and your completely unexplained crippling social anxiety, present despite your perfect and idyllic childhood, curiously appear to present quite similarly to the traumas that have arisen from specific incidents in your life. They color the way you perceive stimuli the same way, they color your instinctive reactions the same way, they color the narratives you construct the same way. A therapist would likely say that you simply haven't gotten to the bottom of it yet - a few more years and tens of thousands of dollars will surely do the trick!
Now, unlike us, the ancient philosophers had no pretense of materialism, of this idea that our minds were formed solely by the atoms and electrical currents in our brains. And so for them, the only rational next step was to search for memories at the root of their deepest imprints, even if nothing was to be found within the bounds of their lifespan. And of course, when some of them happened to discover memories that didn't actually happen, and when processing those memories managed to dissolve their pathologies at last, the only natural conclusion was that some aspect of ourselves, of the stream of impermanent, ever-changing conditions that makes up "us," had previously inhabited other beings.
But, under the framework of materialism, of the laws of physics that seem to explain our world so well, how does any of this make any sense? How can we explain these made-up memories, these visions of past lives, without summarily dismissing the lived experience of our great ancestral teachers?
Perhaps evolution can provide us some answers.
We know that evolution is a sort of greedy algorithm, where existing structures, both mental and physical, are often co-opted through mutations to serve vastly different functions. The bones of the inner ear, previously just normal bones, being co-opted as the most expedient random mutational sequence that ended up performing proprioception is just one canonical example.
We also know that memories, or at least experiences and their resulting mental imprints, are a powerful force, possibly the most powerful force, for shaping the behavior of animals and humans alike. And so if some sort of important behavior (e.g. avoiding snakes) were to be evolved, or even epigenetically passed down, then it makes sense that this process might expediently take the path of co-opting the powerful mental mechanism of memory and imprinting (e.g. a phobia of snakes). That is, by inserting "fake" imprints, "fake" habits into our minds with associated "fake" latent memories at their root, utilizing concepts ("snake", "attacked", "me") that already exist in our circuitry. Indeed, if various innate mental imprints that take on the same form as those created by real experiences could simply be recombined like single nucleotide polymorphisms to provide behavioral diversity to a species, this would be a powerful force for species-level adaptation to ever-changing circumstances. It would also explain why some people end up with rather bizarre recombinations like trypophobia.
And so, not only are our behaviors the recombination of an unending chain of past karmas from uncountably many past lives, so too will the imprints that we obtain from the sensory inputs that make up our current lives recombine to form uncountably many beings in the future. And the state of enlightenment, whether momentary as in the kensho of Zen or permanent as in the nibbana of Theravada, shall be the state in which things are perceived exactly as they are, forming no imprints, no reactionary heuristics, nothing that would make up the infinitely complex recombination of the history of sentient beings that is samsara.
This line of thought can perhaps also clarify the class of concepts variously known as Buddha fields, Pure Lands, astral or causal worlds, heavens, and hells. According to the meditation manuals of the Theravada tradition, a practitioner must first pass through the first four jhanas or meditative states, within which contact with the gross sensory realms drops away, revealing their impermanence and uprooting the identification of the self with them. But an attachment to "fine-material consciousness" still remains, an attachment to the subtle states of bliss, ASMR, rapture, orgasmic sensations that arise in deep meditation. While these are wonderful avenues toward loosening our craving for material attainment, toward providing an alternative peace and bliss untouched by whatever may come through the senses, they are still impermanent just the same, and our attachment to them still causes us to suffer. However, the imprints that cause these attachments are somewhat different in character to usual memories, or even to past life memories; they cannot be identified with any phenomenon in the physical world, or even in the world of thoughts, so subtle is their realization in our perception. And so, for the ancients it was only rational to conclude that these karmas arose from a time when the stream that makes up "us" had abided in a more subtle realm, perhaps a higher realm, one free of the usual fetters of the material world.
And as far as those karmas, those imprints which were so vile so as to appear beyond even the limits of this cruel world's depravity, it was only rational for the ancients to conclude that they were remnants of our time in an even lower world, a hell into which we might still re-descend if we were not careful.